Anomalous Light Scattering by Topological PT -symmetric Particle Arrays: Supplementary Information

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1 Anomalous Light Scattering by Topological PT -symmetric Particle Arrays: Supplementary Information C. W. Ling, Ka Hei Choi, T. C. Mok, Z. Q. Zhang, an Kin Hung Fung, Department of Applie Physics, The Hong Kong Polytechnic University, Hong Kong, China Department of Physics, The Hong Kong University of Science an Technology, Hong Kong, China

2 A. ZAK PHASE AND BAND DISPERSION In Fig. A.a, we show how the Zak phase γ as efine in Eq. 3 changes as the non-hermiticity Im 3 increases. When Im 3 > 0.6, BZ contains broken PT -symmetric phase, an thus γ is not quantize. When Im 3 < 0.6, bans with s = 0.6 are non-trivial, which gives the protecte ege moes integral paths ~hk are shown in Fig. B.. Bulk ban ispersions with Im 3 = 0.05 an 0.5 are emonstrate in Fig. A.b for reference, an in which only bii contains the exceptional points. Note that bulk ispersions for an array with s = 0.4 an s = 0.6 are the same, as the two infinite arrays only iffer in a geometrical shift with /. a i s = 0.4 ii s = 0.6 Regime II Regime I Im b Ban Ban Regime III Regime II Im exceptional points i Im = 0.05 ii Im = 0.5 Plasmon wavevector k FIG. A.. Color online a Zak phases γ with i s = 0.4 an ii s = 0.6. Re 3 =.5. When BZ contains broken PT -symmetric phase regime II, γ is not quantize. Otherwise, it is either classifie as trivial regime I, γ = 0 or non-trivial regime II, γ = π. b Corresponing bulk ispersion relation for s = 0.4 or s = 0.6 when i Im 3 = 0.05 an ii Im 3 = 0.5. B. EXACT ~h-space DIAGRAM FOR PLASMONIC DIMER ARRAYS A close loop is forme by ~hk when k changes from π to π, as shown in Fig. B.. Here, Hk has the form of Eq.. Since the aitional term fk I oes not alter the eigenvectors, Ak share the same eigenvectors an Zak phase γ with Hk. The two eigenvalues are mappe to ω via Eq. 6, which gives the ispersion relation in Fig. A.b. hx s = 0.4 s = 0.6 region with complex eigenvalues hz hy FIG. B.. Color online Numerical integral path ~hk of an infinite plasmonic particle array. Parameters are a = 0.5, b = 0.75, /τ = 0, 3 = i. The region within the kissing cones is where eigenvalues are complex, which correspons to the broken PT -symmetric phase.

3 3 C. ZAK PHASE WITH BI-ORTHONORMAL BASIS If bi-orthonormal bases are use to efine γ instea of the usual efinition, γ will be in general a complex number. However, its real part gives back the value obtaine by Eq. 3. Here, we provie the evaluation of Zak phase base on the bi-orthornormal basis. Recalling H k in Eq. is non-hermitian, therefore, left eigenvectors have to be use to form a bi-orthonormal basis. The left eigenvector u L satisfies the eigenvalue problem H T k ul = E k u L [ 4], whose eigenvalues are E k± = ±h h z, where h = h x + h y. The right an left eigenvectors of H k are [ ] u R hx k ih ± = y k C.a E k± ih z k an u L ± = [ hx k + ih y k E k± ih z k ], C.b where the normalizing factor := h x ih y h x + ih y + E k± ih z = h + E k± ih z = h h z ± ih z h h z. We note that is chosen such that the biorthnormal conitions u L ± u R ± = an u L u R ± = 0 are satisfie. The Zak phase efine by using biorthonormal basis is [,, 4] γ ± = i k u L ± k ur ±, C. π which is similar to Eq. 3. In this case, γ ± will be a complex number even the entire bulk ispersion is in the unbroken PT -symmetric phase, an its real part is the same as that in Eq. 4. To show this, we again restrict E k± are real. We put h x k + ih y k = h ke iφk. Using the prouct rule, Eq. C. becomes The first term in the integrant gives The secon an the forth terms give γ ± =i k π + h h e iφ k + E k± ih z h e iφ k h e iφ M + E k± ih z k± k k h e iφ = ih = ih. k φ + h k φ + h k M + E k± ih z k± = h + E k± ih z = k k M = k± k E k± ih z k k h k, k h. in which we use the efinition of [below Eq. C.]. The thir term gives E k± ih z k E k± ih z = k E k± ih z. C.3 C.4a C.4b C.4c

4 4 Substituting Eqs. C.4 into Eq. C.3, we have γ ± =i π k =i π ih k φ + k h k + ih k k φ + k = k E k± ih z π k k h k φ. C.5 Noticing that as long as E k± are real, h h z are also real. Then, by rationalizing the fraction in the last line of Eq. C.5, we have h h = h h z ± ih h zh h z = h ih h z/h h z h C.6 Finally, by putting Eq. C.6 into Eq. C.5, we have = i h z h h z. γ ± = / π/ = wπ i φ k k i / π/ / π/ h z h k h z h z h k, h z C.7 where w is the wining number of hk about the h z axis. Eq. C.7 shows that γ is in general a complex number, unless hk has some other symmetries so that the integral vanishes.

5 5 D. FIELD PATTERNS FOR A NORMAL ARRAY ɛ 3 =.5 BY MST For comparison with Fig. 5b, here we show the electric fiel pattern of a normal array with ɛ 3 =.5 in Fig. D.. Since the normal array P symmetry, the response of the array shoul be symmetric, which gives an anti-symmetric pattern for the E z fiel component. The two figures Figs. 5 an D. verifie the strong antisymmetry response of the non-hermitian particle array, which oes not contribute to forwar an backwar scattering. FIG. D.. Color online Electric fiel pattern of the E z fiel component at ege moe frequency ω = 0.588ω p of the normal array with ɛ 3 =.5 an s = 0.6. [] J. C. Garrison an E. M. Wright, Complex Geometrical Phases for Dissipative Systems, Phys. Lett. A 8, [] A. I. Nesterov an F. A. e la Cruz, Complex magnetic monopoles, geometric phases an quantum evolution in the vicinity of iabolic an exceptional points, J. Phys. A: Math. Theor. 4, [3] P. Lancaster an M. Tismenetsky, The Theory of Matrices, chap. 4.0, n e. Acaemic Press Inc., Orlano, Floria 3887, 985. [4] A. A. Mailybaev, O. N. Kirillov, an A. P. Seyranian, Geometric phase aroun exceptional points, Phys. Rev. A 7,